There are several. First, use this to add lib paths:
@
a.addLibraryPath("\Program Files\Diagnostico\plugins\sqldrivers\");
@

Then, you are have completely missed the exec() call (a.exec()); Without the event loop (which is started in exec()), signals and slots won't work, the GUI will not work, etc. You need to do all this at a later stage, once the event loop is running. Try redesigning a bit, and ask again if you have troubles. It will be easier to address problems one by one, as they appear.